For a (projective) *smooth* real plane curve $C \subset \mathbb{RP}^2$ the answer is known.

Such a curve is a compact smooth one-dimensional manifold without a boundary, so its connected components are diffeomorphic to circles and are called *ovals*.

If $\deg(C)=d$, a classical result by Harnack shows that the number of ovals cannot exceed $g+1$, where $g=(d-1)(d-2)/2$ is the genus of $C$. Moreover, the bound is sharp: curves with exactly $g+1$ ovals do exist, and are called $M$-*curves*.

For more information, let me refer to the paper

V. I. Arnol’d: Distribution of ovals of the real plane of algebraic curves, of involutions of four-dimensional smooth manifolds, and the arithmetic of integer-valued quadratic forms, *Funct. Anal. Appl.* **5**, 169-176 (1972); translation from *Funkts. Anal. Prilozh*. **5**, No. 3, 1-9 (1971), ZBL0268.53001

that can be downloaded here.